Modeling of Atmospheric Chemistry
Page 3
P
Probability density function
Steric factor
Pe
Péclet number
Pi
Production rate of species i [m–3 s–1]
Associated Legendre polynomial
Pr
Prandtl number
Q
q
Specific humidity [kg water vapor/kg of air]
q
Diabatic heating expressed in K day–1
q
Actinic flux [photons m–2 s–1]
qk
Water concentration in hydrometeor of type k
qλ
Photon flux density [photons m–2 s–1 nm–1]
Q
Diabatic heating rate [J kg–1 s–1 or W m–3]
Qabs
Absorption efficiency
Qext
Extinction efficiency
Qs
Saltation flux [kg m–1 s–1]
Qscat
Scattering efficiency
R
r
Geometric distance from the center of the Earth
r
Position vector
r
Particle radius [m]
rw
Mass mixing ratio of water vapor [kg kg–1]
r
Pearson correlation coefficient
R
Gas constant for air [J K–1 kg–1]
Universal gas constant (8.3143 J K–1 mol–1)
R2
Coefficient of determination
RA
Aerodynamic resistance [s m–1]
RB,i
Boundary resistance for species i [s m–1]
RC,i
Surface resistance for species i [s m–1]
Rd
Gas constant for dry air (287 J K–1 kg–1)
Re
Reynolds number
RH
Relative humidity [percent]
Ri
Richardson number
Ri
Total resistance to dry deposition of species i [s m–1]
RMSE
Root mean square error
Rw
Gas constant for water vapor (461.5 J K–1 kg–1)
S
si
Source rate of species i (in mass) [kg m–3 s–1]
S
Solar energy flux [W m–2] or solar constant (approx. 1368 W m–2)
S
Error covariance matrix
S′
Error correlation matrix
Sa
Aggregation error covariance matrix
SA
Prior error covariance matrix
SI
Instrument error covariance matrix
SM
Forward model error covariance matrix
SO
Observational error covariance matrix
SR
Representation error covariance matrix
Posterior error covariance matrix
Sci
Schmidt number for species i
T
t
Time [s]
t
Student’s variable for the t-test
T
Transmission of radiation
T
Absolute temperature [K]
TE
Effective temperature of the Earth [K]
TKE
Turbulent kinetic energy [m2 s–1]
Ts
Effective temperature of the Sun
Tv
Virtual temperature [K]
U
u
Zonal component of wind velocity [m s–1]
u
Path length [kg m–2]
u*
Friction velocity [m s–1]
u*
Residual zonal wind velocity [m s–1]
uA
Anti-diffusion velocity [m s–1]
ug
Zonal component of the geostrophic wind [m s–1]
u10
Wind velocity 10 m above the surface [m s–1]
V
v
Meridional component of wind velocity [m s–1]
v*
Residual meridional wind velocity [m s–1]
v
Wind velocity vector in Earth’s rotating frame [m s–1]
vg
Meridional component of the geostrophic wind [m s–1]
vi
Mean thermal velocity [m s–1]
V
Molar volume [m3 mol–1]
V
Aerosol volume density [m3 m–3]
V
Wind velocity in inertial frame [m s–1]
VT
Translational Earth’s rotation velocity [m s–1]
W
w
Vertical component of wind velocity [m s–1]
w*
Residual vertical wind velocity [m s–1]
w*
Convective velocity scale [m s–1]
wD,i
Surface deposition velocity of species i [m s–1]
ws
Terminal settling velocity [m s–1]
X
x
Geometric distance in the zonal direction [m]
x
State vector (often refers to the true value)
Optimal estimate of state vector
xA
Prior estimate of state vector
Y
y
Geometric distance in the meridional direction [m]
y
Observation vector
Z
z
Geometric altitude [m]
z0,m
Aerodynamic roughness length [m]
Z
Log pressure altitude [m]
Z
Potential vorticity [s–1 m–1]
ZAB
Collision frequency for molecules A and B [s–1]
α
α
Albedo
α
Aerosol particle size parameter
α
Mass accommodation coefficient
α
Courant number
αT
Thermal diffusion factor
β
β
Fourier number
βext
Aerosol extinction coefficient [m–1]
βabs
Aerosol absorption coefficient [m–1]
βscat
Aerosol scattering coefficient [m–1]
βi,j
Coagulation coefficient for particles i and j [m3 s–1]
γ
γ
Reactive uptake coefficient for heterogeneous chemical process
γ
Regularization factor
γc
Coefficient for non-local turbulent transfer
Γ
Actual atmospheric lapse rate [K m–1]
Γ
Mean age of air [s]
Γd
Dry adiabatic lapse rate [K m–1]
Γw
Wet adiabatic lapse rate [K m–1]
Γϖ
Aggregation matrix
δ
δ
Dirac function
ΔH
Enthalpy of dissolution [J mol–1]
ε
εA
Quantum efficiency (or yield) for the photolysis of molecule A
εO
Observational error vector
εa
Aggregation error vector
εA
Prior estimate error vector
εI
Instrument error vector
εM
Forward model error vector
εR
Representation error vector
ζ
ζ
Relative vorticity of the flow [s–1]
η
η
Step mountain coordinate (eta coordin
ate)
θ
θ
Zenithal direction [radians]
θ
Potential temperature [K]
θ v
Virtual potential temperature [K]
λ
λ
Longitude [radians]
λ
Wavelength [m]
λ
Mean free path of air molecules [m]
λ
Lyapunov exponent [s–1]
λi
Eigenvalue associated with eigenvector ei
Λ
Leaf area index (LAI) [m2 m–2]
μ
μ
Cosine of zenithal direction (θ)
μ
Molecular dynamic viscosity coefficient [Pa s or kg m–1 s–1]
μi
Mass mixing ratio of species i [kg kg–1]
μw
Mass mixing ratio of water vapor [kg kg–1]
ν
ν
Kinematic viscosity [m2 s–1]
ν
Asselin-filter parameter
ν
Frequency [Hz]
νion
Ion-neutral collision frequency [s–1]
π
π
3.14159
ρ
ρa
Mass density of air [kg m–3]
ρd
Mass density of dry air [kg m–3]
ρi
Mass density of species i [kg m–3]
ρp
Mass density of particles or drops [kg m–3]
ρw
Mass density of water vapor [kg m–3]
σ
σ
Stefan-Boltzmann constant (5.67 × 10–8 W m–2 K–4)
σ
Standard deviation
σ
Normalized pressure coordinate (sigma coordinate)
Pseudo density in isentropic coordinates
σA
Absorption cross-section for molecule A [m2]
τ
τ
Optical depth
τ
Lifetime [s]
τ
Stress tensor
τi,j
Element of the stress tensor
φ
φ
Latitude [radians]
φ
Azimuthal direction
ϕ
Radial basis function
Φ
Geopotential [m2 s–2]
Φ∞
Solar flux at the top of the atmosphere [W m–2]
Φk
Basis function in the spectral element method
Φλ
Spectral density of solar flux [W m–2 nm–1]
χ
χ
Solar zenith angle
χ
Velocity potential
ψ
Ψ
Generic mathematical function or variable
Ψ
Streamfunction of the flow
Ψ
Montgomery function (isentropic coordinate system) [J kg–1 or m2 s–2]
ω
ω
“Vertical” velocity in the pressure coordinate system [Pa s–1]
ω
Single scattering albedo
Ω
Angular Earth rotation period (7.292 × 10–5 rad s–1)
Ω
Column concentration [molecules m–2]
Ωs
Slant column concentration [molecules m–2]
1
The Concept of Model
1.1 Introduction
This book describes the foundations of mathematical models for atmospheric chemistry. Atmospheric chemistry is the science that focuses on understanding the factors controlling the chemical composition of the Earth’s atmosphere. Atmospheric chemistry investigates not only chemical processes but also the dynamical processes that drive atmospheric transport, the radiative processes that drive photochemistry and climate forcing, the evolution of aerosol particles and their interactions with clouds, and the exchange with surface reservoirs, including biogeochemical cycling. It is a highly interdisciplinary science.
Atmospheric chemistry is a young and rapidly growing science, motivated by the societal need to understand and predict human perturbations to atmospheric composition. These perturbations have increased greatly over the past century due to population growth, industrialization, and energy demand. They are responsible for a range of environmental problems including degradation of air quality, damage to ecosystems, depletion of stratospheric ozone, and climate change. Quantifying the link between human activities and their atmospheric effects is essential to the development of sound environmental policy.
The three pillars of atmospheric chemistry research are laboratory studies, atmospheric measurements, and models. Laboratory studies uncover and quantify the fundamental chemical processes expected to proceed in the atmosphere. Atmospheric measurements probe the actual system in all of its complexity. Models simulate atmospheric composition using mathematical expressions of the driving physical and chemical processes as informed by the laboratory studies. They can be tested with atmospheric measurements to evaluate and improve current knowledge, and they can be used to make future projections for various scenarios. Models represent a quantitative statement of our current knowledge of atmospheric composition. As such, they are fundamental tools for environmental policy.
Atmospheric chemistry modeling has seen rapid improvement over the past decades, driven by computing resources, improved observations, and demand from policymakers. Thirty years ago, models were so simplified in their treatments of chemistry and transport that they represented little more than conceptual exercises. Today, state-of-science chemical transport models provide realistic descriptions of the 3-D transport and chemical evolution of the atmosphere. Although uncertainties remain large, these models are used extensively to interpret atmospheric observations and to make projections for the future. The state of the science is advancing rapidly, and atmospheric chemists 30 years from now may well scoff at the crude nature of present-day models. Nevertheless, we are now at a point where models can provide a credible, process-based mathematical representation of the atmosphere to serve the needs of science and policy. It is with this perspective of a mature yet evolving state of science that this book endeavors to describe the concepts and algorithms that provide the foundations of atmospheric chemistry models.
This chapter is intended to introduce the reader to the notion and utility of models, and to provide a broad historical perspective on the development of atmospheric chemistry models. It starts with general definitions and properties of mathematical models. It then covers the genesis and evolution of meteorological models, climate models, and finally atmospheric chemistry models, leading to the current state of science. It describes conceptually different types of atmospheric chemistry models and the value of these models as part of atmospheric observing systems. It finishes with a brief overview of the computational hardware that has played a crucial role in the progress of atmospheric modeling.
1.2 What is a Model?
A model is a simplified representation of a complex system that enables inference of the behavior of that system. The Webster New Collegiate Dictionary defines a model as a description or analogy used to help visualize something that cannot be directly observed, or as a system of postulates, data, and inferences presented as a mathematical description of an entity or state of affairs. The Larousse Dictionary defines a model as a formalized structure used to account for an ensemble of phenomena between which certain relations exist. Models are abstractions of reality, and are often associated with the concept of metaphor (Lakoff and Johnson, 1980). Humans constantly create models of the world around them. They observe, analyze, isolate key information, identify variables, establish the relationships between them, and anticipate how these variables will evolve in various scenarios.
One can distinguish between c
ognitive, mathematical, statistical, and laboratory models (Müller and von Storch, 2004). Cognitive models convey ideas and test simple hypotheses without pretending to simulate reality. For example, the Daisyworld model proposed by Lovelock (1989) illustrates the stability of climate through the insolation–vegetation–albedo feedback. This model calculates the changes in the geographical extent of imaginary white and black daisies covering a hypothetical planet in response to changes in the incoming solar energy. It shows that the biosphere can act as a planetary thermostat. Such apparently fanciful models can powerfully illustrate concepts. More formal mathematical models attempt to represent the complex intricacies of real-world systems, and describe the behavior of observed quantities on the basis of known physical, chemical, and biological laws expressed through mathematical equations. They can be tested by comparison to observations and provide predictions of events yet to be experienced. Examples are meteorological models used to perform daily weather forecasts. Statistical models describe the behavior of variables in terms of their observed statistical relationships with other variables, and use these relationships to interpolate or extrapolate behavior. They are empirical in nature, as opposed to the physically based mathematical models. Laboratory models are physical replicas of a system, at a reduced or enlarged geometric scale, used to perform controlled experiments. They mimic the response of the real system to an applied perturbation, and results can be extrapolated to the actual system through appropriate scaling laws.